A mathematical demand function for new Toyotas sold per year for a dealer is as

Question

A mathematical demand function for new Toyotas sold per year for a dealer is as follows:

Qt = 200 – 0.01Pt + 0.005Pm – 10Pg +0.01I + 0.003A

             Qt = quantity purchased,

                Pt = the average price of Toyotas,

                Pm = the average price of Mazdas,

                Pg = the price of gasoline,

                I = per capita income, and

                A = dollars spent annually on advertising

1. Characterize this function by circling all in the following list that are applicable:

univariate, bivariate, multivariate, linear, exponential, logarithmic, curvilinear, 1st degree, 3rd degree, additive, multiplicative, linearly homogeneous

2. What is the numerical value of the partial derivative of the function with respect to advertising spending (be sure to also include the + or – sign. Note: I do not want the symbol for this partial derivative)?

3. Write the mathematical symbol representing the coefficient of the price of gas (the numerical value of this coefficient is -10, but the answer you give is to be the symbol for this partial derivative).

4. Assuming there is a $1 increase in the price of gasoline, what change in Toyota demand will result (give the numerical value, too)?

5. Are Mazadas a substitute for or complementary to Toyotas? What feature of the function tells you?

6. Explain in words what the intercept (which is 200) includes (or does not include).

7. The sign of the variable I is +. Suppose it were negative (that is -.01). What would the negative sign tell you about the type of good Toyotas are?

8. Glory be! Enough pipelines have finally been constructed to carry significantly more oil from the Cushing, OK, petroleum exchange to refineries in Houston. As a result, the price of gasoline decreases by $2.00 per gallon. What change in Toyota demand will result (give the numerical value, too)?

9. The amount budgeted for advertising spending is decreased by $100,000. What change in Toyota demand will result (give the numerical value, too)?

10. Which will cause a greater influence on Toyota demand: a $1,000 decrease in the price of a Toyota or a $1000 increase in the price of a Mazda? Explain what items in the given function give you this information.

Explanation / Answer

1. multivariate, linear

There are more than 3 variables which affect Quantity purchased. It is linear function because there is no exponential in the given function.

2. Qt = 200 – 0.01Pt + 0.005Pm – 10Pg +0.01I + 0.003A

dQt/dA = + 0.003

3. Qt = 200 – 0.01Pt + 0.005Pm – 10Pg +0.01I + 0.003A

dQt/dPg = - 10

4. Qt = 200 – 0.01Pt + 0.005Pm – 10Pg +0.01I + 0.003A

Qt' = 200 – 0.01Pt + 0.005Pm – 10(Pg + 1) +0.01I + 0.003A

Qt' = 200 – 0.01Pt + 0.005Pm – 10Pg - 10 +0.01I + 0.003A

Qt' = Qt - 10

Toyota demand will fall by 10.

5. Qt = 200 – 0.01Pt + 0.005Pm – 10Pg +0.01I + 0.003A

Increase in Pm leads to increase in Qt or toyota demand so these are substitute goods. Positive coefficient of Pm i.e. + 0.005 shows direct relation between Pm and Qt.

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